# Post-Selection Inference for Generalized Linear Models with Many Controls

arXiv: https://arxiv.org/abs/1304.3969 · HTML: https://mlsanity.com/paper/1304.3969/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 15 Apr 2013.

## Built on by (most cited first, 11 of 11)

- [Semi-parametric inference based on adaptively collected data](https://mlsanity.com/paper/2303.02534/) (2023)
- [Statistical Limits of Adaptive Linear Models: Low-Dimensional Estimation and Inference](https://mlsanity.com/paper/2310.00532/) (2023)
- [CATE Lasso: Conditional Average Treatment Effect Estimation with High-Dimensional Linear Regression](https://mlsanity.com/paper/2310.16819/) (2023)
- [Adaptive debiased SGD in high-dimensional GLMs with streaming data](https://mlsanity.com/paper/2405.18284/) (2024)
- [U-learning for Prediction Inference via Combinatory Multi-Subsampling: With Applications to LASSO and Neural Networks](https://mlsanity.com/paper/2407.15301/) (2024)
- [Generalized Linear Models with Structured Sparsity Estimators](https://mlsanity.com/paper/2104.14371/) (2021)
- [An Overview of Healthcare Data Analytics With Applications to the COVID-19 Pandemic](https://mlsanity.com/paper/2111.14623/) (2021)
- [Average Adjusted Association: Efficient Estimation with High Dimensional Confounders](https://mlsanity.com/paper/2205.14048/) (2022)
- [Penalized Likelihood Inference with Survey Data](https://mlsanity.com/paper/2304.07855/) (2023)
- [Triple/Debiased Lasso for Statistical Inference of Conditional Average Treatment Effects](https://mlsanity.com/paper/2403.03240/) (2024)
- [The Post Double LASSO for Efficiency Analysis](https://mlsanity.com/paper/2505.14282/) (2025)

## Abstract (arXiv)

This paper considers generalized linear models in the presence of many controls. We lay out a general methodology to estimate an effect of interest based on the construction of an instrument that immunize against model selection mistakes and apply it to the case of logistic binary choice model. More specifically we propose new methods for estimating and constructing confidence regions for a regression parameter of primary interest α0, a parameter in front of the regressor of interest, such as the treatment variable or a policy variable. These methods allow to estimate α0 at the root-n rate when the total number p of other regressors, called controls, potentially exceed the sample size n using sparsity assumptions. The sparsity assumption means that there is a subset of s<n controls which suffices to accurately approximate the nuisance part of the regression function. Importantly, the estimators and these resulting confidence regions are valid uniformly over s-sparse models satisfying s2log2 p = o(n) and other technical conditions. These procedures do not rely on traditional consistent model selection arguments for their validity. In fact, they are robust with respect to moderate model selection mistakes in variable selection. Under suitable conditions, the estimators are semi-parametrically efficient in the sense of attaining the semi-parametric efficiency bounds for the class of models in this paper.